Lesson 1 Growing Roots Develop Understanding

Learning Focus

Make observations about the domain, range, and rate of change of the square root function.

How does the square root function, $f (x) = \sqrt{x}$ , grow?

Open Up the Math: Launch, Explore, Discuss

In your previous math work with the Pythagorean theorem, you have learned about square roots as numbers. In this task, we will consider the square root function. We will begin by examining the context that gave rise to the name of this function, the relationship between the area and side length of a square.

1.

Determine the area and side length of each of the following squares and label both quantities on the diagram. Illustrate or explain how you found the measure of each quantity.

You may recall the following definition of square root: The square root of a number, $N$ , is the factor, $n$ , such that when $n$ is multiplied by itself, $n \times n$ , the product is $N$ .

2.

Describe the following key features of the square root function, $f (x) = \sqrt{x}$ .

Justify each of your claims by referencing the definition of square root, considering the context of the area and side length of a square or using your sense of numbers and operations.

a.

Intervals where $f (x) = \sqrt{x}$ increases or decreases (Hint: As $x$ increases, what might we say about the change in $\sqrt{x}$ ?)

How do you know?

b.

Maximum and minimum values of $f (x) = \sqrt{x}$

How do you know?

c.

Intercepts for the graph of $f (x) = \sqrt{x}$

How do you know?

d.

Domain and range of $f (x) = \sqrt{x}$

How do you know?

Consider the hint, “As $x$ increases, what might we say about the change in $\sqrt{x}$ ?” This hint probably led you to say something like, “As $x$ increases, $\sqrt{x}$ increases since larger numbers would require larger square root factors.” You might have justified this by considering the area of a square—the larger the area, the longer the side length.

What can we say about the way the square root function increases? Does it increase at a constant rate, at an increasing rate, or at a decreasing rate? It may be hard to tell by just thinking about squares with larger and larger area. The following exploration will help you develop some sense about how the square root function grows.

The diagram shown was created using the following recursive process:

• Step 1: Construct the first line segment from $(0, 0)$ to $(1, 0)$ on the coordinate grid.

• Recursive Rule: Construct the $n^{t h}$ line segment as the hypotenuse of the right triangle formed using the ${(n - 1)}^{t h}$ line segment as one of the legs and the other leg being a line segment of $1$ unit in length drawn perpendicular to the endpoint farthest from the origin of the ${(n - 1)}^{t h}$ line segment, so that the sequence of line segments spiral-out counterclockwise around the origin.

3.

Determine the length of each line segment in this sequence, measured from the origin to the endpoint labeled in the diagram:

Segment Endpoint	Length
A	$1$
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P

4.

Let $f (n)$ be the numerical length of the $n th$ line segment in this sequence.

a.

Define $f (n)$ recursively: $f (1) = 1$ , and $f (n) =$

b.

Define $f (n)$ explicitly: $f (n) =$

5.

Use a compass to draw circles on the diagram, centered at the origin with radii of $1$ , $2$ , $3$ , and $4$ units.

a.

What is significant about the endpoints in the diagram that lie on these circles?

b.

If we were to continue to construct line segments using this recursive pattern, how many endpoints would lie between the circle of radius $4$ and the circle of radius $5$ ?

c.

What observations can you make about the number of endpoints that lie between consecutive circles?

The domain of this discrete sequence is the natural numbers. To graph this sequence, we could use the lengths of the line segments in the diagram to determine the value of the sequence for each natural number, as shown in the graph.

If we don’t restrict the domain of this function to the natural numbers, we create the continuous square root function, $f (x) = \sqrt{x}$ , which you described in problem 1.

6.

How does the square root function grow?

Ready for More?

Prove algebraically the generalization that was observed in problem 4c. That is, prove that between the $n^{t h}$ and the ${(n + 1)}^{t h}$ circle in the diagram, there will be $2 n$ endpoints. (Another way to say this is that between the $n^{t h}$ and the ${(n + 1)}^{t h}$ perfect square, there are $2 n$ integers that are not perfect squares.)

Takeaways

The square root function, $f (x) = \sqrt{x}$ , has the following key features:

Domain:

Range:

Intercepts:

Minimum value:

Maximum value:

Continuous or discrete:

Rate of change:

Vocabulary

square root function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we extended our understanding of square roots by defining the square root function, $f (x) = \sqrt{x}$ , and exploring the key features of its graph.

Retrieval

Identify whether the given equation represents a direct variation function. If it does, state the constant of variation. If the equation does not represent a direct variation function, explain why not.

1.

$8 x + 5 y = 0$

2.

$2 x - 5 y = 1$

3.

Describe the transformation on the parabola. Then graph the function.

$y = \frac{1}{4} {(x + 3)}^{2} - 6$